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Chapter 9: Problem 27

Force \(F_{x}=(10 \mathrm{N}) \sin (2 \pi t / 4.0 \mathrm{s})\) is exerted on a \(250 \mathrm{g}\) parti- cle during the interval 0 s \(\leq t \leq 2.0\) s. If the particle starts from rest, what is its speed at \(t=2.0 \mathrm{s} ?\)

Short Answer

Expert verified
The final speed of the particle at t=2.0s is \(80 m/s\).

Step by step solution

01

Calculate the Impulse

Impulse is defined as the change in momentum that a particle experiences due to a force applied on it. Mathematically, impulse can be defined as the time integral of the force applied. The force applied on the particle is \(F_x = (10N) * \sin(2蟺t/4.0s)\) and this force is applied from t=0s to t=2.0s. So, the impulse can be calculated as \(J = 鈭玙{0 s}^{2.0 s} F_x dt\).
02

Evaluating the Integral

To evaluate this integral, we have to integrate the function \(F_x = (10N) * \sin(2蟺t/4.0s)\) from 0 to 2. Upon integrating, we get \(J = (10N * 4.0s / 2蟺) * [-cos(2蟺t/4.0s)]_{0 s}^{2.0 s}\). Simplifying this yields \(J = 20 N*s\).
03

Calculating the Final Momentum

The final momentum of the particle can be calculated by adding the impulse to the initial momentum. As per the problem, the particle starts from rest, so, its initial momentum will be zero. Therefore, the final momentum \(p_f\) of the particle will be equal to the impulse \(J\) exerted on it. Thus, \(p_f = J = 20 N*s\).
04

Calculating the Final Speed

The speed of the particle can be calculated by dividing the final momentum by the mass of the particle. So, the final speed \(v_f\) can be calculated as \(v_f = p_f / m = 20 N*s / 0.250 kg\). Simplifying, we find that \(v_f = 80 m/s\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Impulse and Momentum
When studying classical mechanics, impulse and momentum are crucial concepts that describe the motion of objects when forces are applied. Let's break them down in simple terms.

Impulse is the effect of a force applied over a period of time that changes the momentum of an object.
  • The impulse experienced by an object is equivalent to the change in its momentum.
  • It is calculated using the integral over time of the force applied: \( J = \int F dt \).
  • In practical terms, this means that a greater force or a longer time period results in a larger impulse.
Momentum, on the other hand, is the quantity of motion an object has. It is influenced by both the object's mass and speed.
  • Momentum (\( p \)) is calculated as the product of an object鈥檚 mass (\( m \)) and its velocity (\( v \)): \( p = m \cdot v \).
  • In a system where no external forces act, momentum is conserved. This means the total momentum remains constant before and after any event.
In the given exercise, we used impulse to find the final momentum because the particle starts from rest. Thus, the impulse equals the total change in momentum.
Force and Motion
Force is a push or pull that can cause an object to accelerate, slow down, stay in place, or change shape. The relationship between force and motion is a fundamental aspect of classical mechanics, often described by Newton's laws.
  • Force (\( F \)) can cause a change in an object's velocity depending on its mass (\( m \)).
  • The basic equation governing this relationship is \( F = m \cdot a \), where \( a \) is acceleration.
  • A force that varies over time can result in a complex motion, which often requires calculus for analysis.
In our exercise problem, a time-varying force \( F_x = (10 N) \sin\left(\frac{2\pi t}{4.0 s}\right) \) is considered. Such a sinusoidal force suggests that the force changes periodically over time, impacting the particle鈥檚 motion through its effect on the momentum. This variation necessitates integration to calculate the impulse, fundamentally linking force and motion.
Calculus in Physics
In physics, calculus is an essential tool used to model and solve problems involving rates of change and areas under curves, among other concepts. Here鈥檚 how calculus fits into the problem at hand.

Calculating impulse involves integrating the force function over a specific time interval.
  • Integration is the process of finding the area under the curve of a function, which in this context gives the total impulse exerted on the particle over time.
  • For our problem, we integrate \( F_x(t) = (10 N) \sin\left(\frac{2\pi t}{4.0 s}\right) \) from \( t = 0 \) to \( t = 2.0 \) seconds.
  • This process results in an impulse value, which we connect with momentum to determine the final velocity of the particle.
Differentiation, the inverse operation of integration, also plays a key role in physics, particularly in understanding how velocity and acceleration relate to position over time.
  • In problems where forces are differentiated, finding the rate of change of velocity (acceleration) and thereby force analysis becomes essential.
Therefore, the use of calculus allows physicists to precisely model the complex behavior of forces and motions over continuous time periods.

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Most popular questions from this chapter

A \(1000 \mathrm{kg}\) cart is rolling to the right at \(5.0 \mathrm{m} / \mathrm{s} .\) A \(70 \mathrm{kg}\) man is standing on the right end of the cart. What is the speed of the cart if the man suddenly starts running to the left with a speed of \(10 \mathrm{m} / \mathrm{s}\) relative to the cart?

Two ice skaters, with masses of \(50 \mathrm{kg}\) and \(75 \mathrm{kg}\), are at the center of a \(60-\) m-diameter circular rink. The skaters push off against each other and glide to opposite edges of the rink. If the heavier skater reaches the edge in \(20 \mathrm{s}\), how long does the lighter skater take to reach the edge?

The Army of the Nation of Whynot has a plan to propel small vehicles across the battlefield by shooting them, from behind, with a machine gun. They've hired you as a consultant to help with an upcoming test. A \(100 \mathrm{kg}\) test vehicle will roll along frictionless rails. The vehicle has a tall "sail" made of ultrahard steel. Previous tests have shown that a \(20 \mathrm{g}\) bullet traveling at \(400 \mathrm{m} / \mathrm{s}\) rebounds from the sail at \(200 \mathrm{m} / \mathrm{s}\). The design objective is for the cart to reach a speed of \(12 \mathrm{m} / \mathrm{s}\) in \(20 \mathrm{s}\). You need to tell them how many bullets to fire per second. A five-star general will watch the test, so your ability to get future consulting jobs will depend on the outcome.

A 50kg archer, standing on frictionless ice, shoots a \(100 \mathrm{g}\) arrow at a speed of \(100 \mathrm{m} / \mathrm{s}\). What is the recoil speed of the archer?

\(A 10,000 \mathrm{kg}\) railroad car is rolling at \(2.0 \mathrm{m} / \mathrm{s}\) when a \(4000 \mathrm{kg}\) load of gravel is suddenly dropped in. What is the car's speed just after the gravel is loaded?
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